Existence and multiplicity of periodic solutions to differential equations with attractive singularities

The existence and multiplicity of T-periodic solutions to a class of differential equations with attractive singularities at the origin are investigated in the paper. The approach is based on a new method of construction of strict upper and lower functions. The multiplicity results of Ambrosetti–Prodi type are established using a priori estimates and certain properties of topological degree.

COLLOCATION METHOD FOR FUZZY VOLTERRA INTEGRAL EQUATIONS OF THE SECOND KIND

In this paper we consider fuzzy Volterra integral equation of the second kind whose kernel may change sign. We give conditions for smoothness of the upper and lower functions of the solution. For numerical solution we propose the collocation method with two different basis function sets: triangular and rectangular basis. The smoothness results allow us to obtain the convergence rates of the methods. The proposed methods are illustrated by numerical examples, which confirm the theoretical convergence estimates.

ON CONSTRUCTION OF CONVERGING SEQUENCES TO SOLUTIONS OF BOUNDARY VALUE PROBLEMS

We consider the Dirichlet problem x″ = f(t,x), x(a) = A, x(b) = B under the assumption that there exist the upper and lower functions. We distinguish between two types of solutions, the first one, which can be approximated by monotone sequences of solutions (the so called Jackson—Schrader's solutions) and those solutions of the problem, which cannot be approximated by monotone sequences. We discuss the conditions under which this second type solutions of the Dirichlet problem can be approximated.